\documentclass{article}
\usepackage{ifthen}
\usepackage{epsfig}
\usepackage[utf8]{inputenc}
% Packages requested by user
\usepackage{amsmath}

\usepackage{newunicodechar}
  \newunicodechar{⁻}{${}^{-}$}% Superscript minus
  \newunicodechar{²}{${}^{2}$}% Superscript two
  \newunicodechar{³}{${}^{3}$}% Superscript three

\pagestyle{empty}
\begin{document}
$x$
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$y$
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$\dot{x}$
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$\dot{y}$
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\[x_k = f(x_{k-1}, u_{k-1}) + v_{k-1}\]
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\[z_k = h(x_k) + w_k\]
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$f$
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$h$
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$v$
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$w$
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$z$
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$u$
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$x_k$
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$z_k$
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$\hat{x}_k$
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$\hat{z}_k$
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\[y = \begin{bmatrix} f_1(x_1,x_2,x_3) \\ f_2(x_1,x_2,x_3) \end{bmatrix} \]
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\[Jac(y)=\begin{bmatrix} \frac{\partial f_1}{\partial x_1} \frac{\partial f_1}{\partial x_2} \frac{\partial f_1}{\partial x_3} \\ \frac{\partial f_2}{\partial x_1} \frac{\partial f_2}{\partial x_2} \frac{\partial f_2}{\partial x_3} \end{bmatrix} \]
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\[Hess(y(1)) =\begin{bmatrix} \frac{\partial^2 f_1}{\partial x_1^2} \frac{\partial^2 f_1}{\partial x_1 \partial x_2} \frac{\partial^2 f_1}{\partial x_1 \partial x_3} \\ \frac{\partial^2 f_1}{\partial x_2 \partial x_1} \frac{\partial^2 f_1}{\partial x_2^2} \frac{\partial^2 f_1}{\partial x_2 \partial x_3} \\ \frac{\partial^2 f_1}{\partial x_3 \partial x_1} \frac{\partial^2 f_1}{\partial x_3 \partial x_2} \frac{\partial^2 f_1}{\partial x_3^2} \end{bmatrix} \]
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\[Hess(y(2)) =\begin{bmatrix} \frac{\partial^2 f_2}{\partial x_1^2} \frac{\partial^2 f_2}{\partial x_1 \partial x_2} \frac{\partial^2 f_2}{\partial x_1 \partial x_3} \\ \frac{\partial^2 f_2}{\partial x_2 \partial x_1} \frac{\partial^2 f_2}{\partial x_2^2} \frac{\partial^2 f_2}{\partial x_2 \partial x_3} \\ \frac{\partial^2 f_2}{\partial x_3 \partial x_1} \frac{\partial^2 f_2}{\partial x_3 \partial x_2} \frac{\partial^2 f_2}{\partial x_3^2} \end{bmatrix} \]
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\[x_k = Ax_{k-1} + Bu_{k-1} + v_{k-1}\]
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\[z_k = Hx_k + w_k\]
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$A$
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$B$
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$H$
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\[Q = Q^T\]
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\[R = R^T\]
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\[x_k = 12 + v_{k-1}\]
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\[z_k = x_k + w_ḳ\]
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$A = 0$
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$B = 1$
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$H = 1$
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$u = 12$
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$x_0 = 12$
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$Q=4$
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$R=4$
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$Q$
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$R$
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\[ f = \begin{bmatrix} \sin(x_2(k-1))(k-1) \\ x_2(k-1) \end{bmatrix} \]
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\[ h = \begin{bmatrix} x_1(k) \\ x_2(k) \end{bmatrix} \]
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\[ x_0 = \begin{bmatrix} 0 \\ \frac{1 \pi}{500} \end{bmatrix}\]
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\[ f = \begin{bmatrix} x_2(k) \\ x_3(k) \\ 0.005 \, x_1(k) \bigg(x_2(k) + x_3(k) \bigg) \end{bmatrix} \]
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\[ h = \begin{bmatrix} x_1(k) \end{bmatrix} \]
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\[ x_0 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\]
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$x_1$
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$x_2$
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$x_3$
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$u_k$
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\[x_k = x_{k-1} + v_{k-1}\]
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\end{document}
